**1. **Fermentation Kinetics Alfred Carlson

**2. **Fermentation Kinetics There are 4 things to understand about cell growth
Cell growth is exponential
The specific growth rate depends on environmental conditions
An empirical equation called the Monod equation is used to relate specific growth rate to the substrate concentration in the media
Production of products

**3. **Scenario A 1 L reactor containing 2g/L of glucose in a minimal media was inoculated with 1 cc of an overnight shake flask culture. The OD of the shake culture was measured to be 7. For the next 16 hours, samples were taken from the reactor every 2-3 hours and the OD was measured in a spectrophotometer @ 600 nm. The results are on the next slide.

**4. **Bioreactor Sampling for OD

**5. **Results Time (hr) OD reading
0 ND (not determined)
2 0.007
4 0.02
7 0.06
10 0.18
12.5 0.45
15 1.45

**6. **What you do with this data The ?specific growth rate? is a parameter that characterizes the cell growth and is another way of measuring the ?doubling time? If the cells are growing right, they will have a constant specific growth rate that falls in a ?normal range?.

**7. **Steps - Plot Plot the data on a semilog plot of (log) OD vs t ? this gives you a big picture view of the data.

**8. **Steps - Regress 2. Fit the data using an exponential growth model
OD = ODoexp{mt}

**9. **Fitting to Monod model Use Monod, substrate limited kinetics to model the growth of these cells. Predict the entire growth curve and predict the glucose depletion curve for the cells based on a yield factor of 0.5 gcells/gram glucose.

**10. **Monod model The Monod model accounts for the fact that the cells specific growth rate decreases (doubling time increases) when the concentration of food is low. This is the most common growth model.
m = mmax*S/(Ks+S)

**11. **Using the Monod model Either you have to solve 2 simultaneous differential equations, or you have to do periodic update material balances:
Cell balance = dX/dt = mX
Substrate balance
dS/dt = -mX/Yx/s
Remember: m = mmax*S/(Ks+S)

**12. **Update method

**13. **Model graph

**14. **Glucose depletion

**15. **Lecture slides Doubling time vs specific growth rate
Temperature effect on specific growth rate
Ribosome limited growth
Product expression models

**16. **Cell Doubling Cells grow by doubling at regular intervals

**17. **Culture Kinetics Exponential Growth
Specific growth rate
N DN/Dt
1 1
2 2
4 4
8 8

**18. **Exponential growth As long as the doubling time remains constant, the specific growth rate (m) will remain constant.

**19. **N, Cell mass, OD For constant volume reators ? The number concentration of cells n (cells/L) is proportional to N (n = N/V). The cell mass is proportional to N, and OD is proportional to n. ?
All measurements of cell growth follow exponential law.

**20. **Cell Growth Data

**21. **Doubling Time and Specific Growth Rate The specific growth rate is just another way of expressing the doubling time.
During a doubling time N/No = 2
Ln(N/No) = Ln(2) = 0.693 = mtd
td = 0.693/m

**22. **Culture Kinetics Cell type doubling time growth rate (hr-1)
bacteria 20 ? 90 min ~0.3 - 2.1
yeast 45-120 min ~0.2 ? 1.0
mold 4-8 hours ~0.1-0.2
mammalian cells 20-48 hours ~0.05
These are just rough numbers, not rules.

**23. **Example A certain yeast cell is said to have a doubling time of 90 minutes.
What is its specific growth rate in hr-1?
A 1 cc inoculum of the same cells at an OD of 5 (600 nm) is used to inoculate 1 L of fresh media.
How long will it take for the new culture to reach an OD of 1? of 5?

**24. **Answers A certain yeast cell is said to have a doubling time of 90 minutes. What is its specific growth rate in hr-1?
m = 0.693/td
td = 1.5 hr ? m = 0.693/1.5 = 0.462 hr-1

**25. **Answers A 1 cc inoculum of the same cells at an OD of 5 (600 nm) is used to inoculate 1 L of fresh media. How long will it take for the new culture to reach an OD of 1? Of 5?
Ln (OD/ODo) = mt ODo = 5/1001 ~ 0.005
t1 = ln (1/.005)/0.462 = 11.5 hours
t5 = ln (5/.005)/0.462 = 14.95 hours

**26. **Environmental Conditions The specific growth rate is the fastest the cells can double on that particular media at that particular temperature.

**27. **Growth vs Temp

**28. **pH effect

**29. **pH effect Optimum growth below 4 ? acidophiles
Optimum growth between 6 and 8 ? mesophiles
Optimum growth above 8 - alkaliphiles

**30. **Maximum growth rate determined by ribosomes In order for cells to double regularly they need to duplicate their duplication system in the doubling time.
Only 16 amino acids per second can be added to a growing protein chain (ribosome limit)

**31. **Limits to Cell Growth Rate For cells to grow they must reproduce themselves
This requires (minimum) the reproduction of the reproduction system (PSS)

**32. **Limits to Cell Growth Rate Translation is rate limiting
Time required for ribosome self-reproduction is minimum doubling time

**33. **Limits to Cell Growth Rate Reproduction time =
Number of amino acids per ribosome
Rate of amino acid addition to protein chain

**34. **Limits to Cell Growth Rate 10,000 amino acids/ ribosome
15 -20 amino acids added per second
~ 600 seconds (10 minutes) to reproduce a ribosome
10 minutes for ribosomes to reproduce ribosomes

**35. **Limits to Cell Growth Rate Production of other proteins slow down ribosome reproduction
minimum need is 20,000 aa/ribosome = 20 minute doubling time (observed = 17 minutes or so)
More complex cells = more aa addition = slower growth

**36. **Limits to Cell Growth Rate On poor media cells must make more proteins for getting and processing food ---- slower growth
When cells produce product, more protein is needed for these processes ---- slower growth

**37. **Limits to Cell Growth Rate Net effect of engineering on cells (ideal)
Ribosomes diverted from required functions
growth rate cut proportional to 2 x fraction of foreign protein produced -- at 50% protein growth rate = 0

**38. **Cell Growth Modeling When there is no substrate, cells can?t double and m goes to zero
No matter how much substrate, cells can only double so fast (m goes to mmax)

**39. **Monod Equation Growth equation (Monod equation)

**40. **Stationary Phase Growth equations with substrate exhaustion:
Cells: (rate of change of cell concentration)
dX/dt = mX = mmaxS/(Ks + S) * X
Substrate: (rate of change of substrate conc)
dS/dt = -1/Yx/S dX/dt = -1/Yx/S mmaxS/(Ks + S) X

**41. **Substrate depletion in batch culture

**42. **Batch Reactor Kinetics ? Real data

**43. **Cell growth kinetics

**44. **Batch Reactor Kinetics

**45. **Products Metabolites (stoichiometric)
Example: how much ethanol can a cell make from a gram of glucose: (Facts ? hardly any cell mass therefore follow glycolytic pathway to ethanol.)( 2 moles per mole glucose) 1 gram = .0055 moles ? .0110 moles ethanol x 46 g/mole = 0.5 grams. The rest is CO2.
Rate of ethanol production = rate of glucose consumption x 2

**46. **Products Non-metabolites (2 broad types)
Growth associated products
Example: Cell components: Cells must be growing to produce product.
dP/dt = b dX/dt = bmX

**47. **Non-metabolites (2 broad types)
Non-Growth associated products
Example: Penicillin
dP/dt = aX
Products

**48. **Overall Production Coefficient Production coefficient
dP/dt = (bm + a) X
dP/dt = qpX

**49. **Summary Cell growth kinetics are described by either the doubling time or the specific growth rate.
Monod kinetics take into account that cells stop growing when the substrate runs out.
Ultimately cells grow at a certain rate because of what the ribosomes are doing
Product expression kinetics are either related to the consumption rate of precursors or to empirical patterns of expression called growth related or non-growth related expression